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Theorem elmpps 35055
Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p 𝑃 = (mPreSt‘𝑇)
mppsval.j 𝐽 = (mPPSt‘𝑇)
mppsval.c 𝐶 = (mCls‘𝑇)
Assertion
Ref Expression
elmpps (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ot 4630 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
2 mppsval.p . . . 4 𝑃 = (mPreSt‘𝑇)
3 mppsval.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsval.c . . . 4 𝐶 = (mCls‘𝑇)
52, 3, 4mppsval 35054 . . 3 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
61, 5eleq12i 2818 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
7 oprabss 7508 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ ((V × V) × V)
87sseli 3971 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
92mpstssv 35021 . . . . . 6 𝑃 ⊆ ((V × V) × V)
109sseli 3971 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
111, 10eqeltrrid 2830 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
1211adantr 480 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13 opelxp 5703 . . . 4 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14 opelxp 5703 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15 simp1 1133 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑑 = 𝐷)
16 simp2 1134 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → = 𝐻)
17 simp3 1135 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑎 = 𝐴)
1815, 16, 17oteq123d 4881 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ⟨𝑑, , 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1918eleq1d 2810 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (⟨𝑑, , 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
2015, 16oveq12d 7420 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑑𝐶) = (𝐷𝐶𝐻))
2117, 20eleq12d 2819 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
2219, 21anbi12d 630 . . . . . . 7 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ((⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2322eloprabga 7509 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
24233expa 1115 . . . . 5 (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2514, 24sylanb 580 . . . 4 ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2613, 25sylbi 216 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
278, 12, 26pm5.21nii 378 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
286, 27bitri 275 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  cop 4627  cotp 4629   × cxp 5665  cfv 6534  (class class class)co 7402  {coprab 7403  mPreStcmpst 34955  mClscmcls 34959  mPPStcmpps 34960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpst 34975  df-mpps 34980
This theorem is referenced by:  mthmpps  35064  mclspps  35066
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